# transformations replaced by symmetry

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## 8 matching pages

##### 1: 19.15 Advantages of Symmetry
Symmetry in $x,y,z$ of $R_{F}\left(x,y,z\right)$, $R_{G}\left(x,y,z\right)$, and $R_{J}\left(x,y,z,p\right)$ replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17). … …
##### 3: 19.25 Relations to Other Functions
19.25.16 $\Pi\left(\phi,\alpha^{2},k\right)=-\tfrac{1}{3}\omega^{2}R_{J}\left(c-1,c-k^{2% },c,c-\omega^{2}\right)+\sqrt{\frac{(c-1)(c-k^{2})}{(\alpha^{2}-1)(1-\omega^{2% })}}\*R_{C}\left(c(\alpha^{2}-1)(1-\omega^{2}),(\alpha^{2}-c)(c-\omega^{2})% \right),$ $\omega^{2}=k^{2}/\alpha^{2}$.
##### 4: Bille C. Carlson
The main theme of Carlson’s mathematical research has been to expose previously hidden permutation symmetries that can eliminate a set of transformations and thereby replace many formulas by a few. …
##### 5: Errata
• Chapter 18 Orthogonal Polynomials

The reference Ismail (2005) has been replaced throughout by the further corrected paperback version Ismail (2009).

• Figure 4.3.1

This figure was rescaled, with symmetry lines added, to make evident the symmetry due to the inverse relationship between the two functions.

Reported 2015-11-12 by James W. Pitman.

• Section 27.20

The entire Section was replaced.

• Subsection 21.10(i)

The entire original content of this subsection has been replaced by a reference.

• Equations (18.16.12), (18.16.13)

The upper and lower bounds given have been replaced with stronger bounds.

• ##### 6: 18.7 Interrelations and Limit Relations
###### §18.7(ii) Quadratic Transformations
18.7.17 $U_{2n}\left(x\right)=W_{n}\left(2x^{2}-1\right),$
18.7.18 $T_{2n+1}\left(x\right)=xV_{n}\left(2x^{2}-1\right).$
##### 7: 18.17 Integrals
and three formulas similar to (18.17.9)–(18.17.11) by symmetry; compare the second row in Table 18.6.1. …
##### 8: 20.11 Generalizations and Analogs
If both $m,n$ are positive, then $G(m,n)$ allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): …
###### §20.11(v) Permutation Symmetry
The importance of these combined theta functions is that sets of twelve equations for the theta functions often can be replaced by corresponding sets of three equations of the combined theta functions, plus permutation symmetry. …